Minimally saturated models
A model for a complete first order theory T in a language of finite type is minimally saturated if it is recursively saturated and elementarily embeddable in every recursively saturated model of T. Such a model is unique when it exists, and may be regarded as the smallest model of T with saturation properties. (Alternatively, if T* denotes the theory obtained from T by adding all Σ 1 1 -sentences consistent with T, then a minimally saturated model for T is simply a prime model for T*). We show that the existence of such a model is implied by the existence of a countably saturated model, and in turn implies the existence of a prime model, and that both these implications are strict. We also give an easily applicable sufficient condition for T to have no minimally saturated model. §3 includes a general result about the degrees of complete types of first-order theories.
Unable to display preview. Download preview PDF.
- J. Barwise and J. Schlipf, An Introduction to Recursively Saturated and Resplendent Models, J.S.L. 41, 1976.Google Scholar
- C.C. Chang and H.J. Keisler, Model Theory, North Holland, 1973.Google Scholar
- H. Friedman, Countable Models of Set Theories, Cambridge Summer School in Mathematical Logic, 1971. Springer Lecture Notes 337 pp. 539–573.Google Scholar
- T.J. Grilliot, Omitting Types: Application to Recursive Theory, JSL 37 No. 1. pp.81–89.Google Scholar
- 72.C.G. Jockusch, and R.I. Soare, π10-classes and degrees of theories, Trans. AMS 173 Nov. 1972 pp. 33–56.Google Scholar
- H. Lessan, Ph.D. Thesis, Manchester 1978Google Scholar
- H. Lessan and G.M. Wilmers, Scott sets and non-standard model theory (to appear).Google Scholar
- A. MacIntyre, Decidable Recursively Saturated Models (to appear).Google Scholar
- D. Scott, Algebras of sets binumerable in complete extensions of arithmetic, Proc. Sympos. Pure Math, Vol. 5, AMS, 1962.Google Scholar
- G.M. Wilmers, Ph.D. Thesis, Oxford 1975.Google Scholar